differential topology - Is there a Morse theory for sections of bundles or more generally for maps?

Ryan: my use of the phrasing 'classify functions' is both inaccurate and imprecise. But we're in danger of talking past one another, because the classification I am pursuing is on the basis of different physical processes. This is a physics goal, not a mathematical one.

The functions observational cosmologists consider are invariably smooth. The cosmological dark matter density is an example and the calculation of the Euler characteristic for level sets of that function is a nice application of Morse theory to physics. The velocity of the dark matter fluid, which a physicist would call a vector-valued function, is an example of an object to which we would like to apply the same apparatus, but are unsure how to proceed.

I think that you indicate that the concept of a level set is not well-formed for an object of that kind? Could you comment on the generality of that statement bearing in mind the relative triviality of the objects in question?

I would also like to clarify Jose's phrasing. Studies of this kind aim very much to study the properties of $f$, rather than $M$. The study of the topology of the spatial Universe is an excellent and interesting problem as well! But the aim here is to use techniques from Morse theory as a way of probing how physical effects alter the form of $f$.

Thanks for all the help so far! I hope there's more to discuss.